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Abstract

It is well known that the addition of noise to a multistable dynamical system can induce random transitions from
one stable state to another. For low noise, the times between transitions have an exponential tail and Kramers’
formula gives an expression for the mean escape time in the asymptotic limit. If a number of multistable systems
are coupled into a network structure, a transition at one site may change the transition properties at other sites.
We study the case of escape from a “quiescent” attractor to an “active” attractor in which transitions back
can be ignored. There are qualitatively different regimes of transition, depending on coupling strength. For
small coupling strengths, the transition rates are simply modified but the transitions remain stochastic. For large
coupling strengths, transitions happen approximately in synchrony—we call this a “fast domino” regime. There
is also an intermediate coupling regime where some transitions happen inexorably but with a delay that may be
arbitrarily long—we call this a “slow domino” regime. We characterize these regimes in the low noise limit in
terms of bifurcations of the potential landscape of a coupled system. We demonstrate the effect of the coupling
on the distribution of timings and (in general) the sequences of escapes of the system.

Funders/Sponsor

The authors gratefully acknowledge the financial support
of the EPSRC via Grant No. EP/N014391/1. We
thank the anonymous referees for their comments, criticisms,
and suggestions. P.A. gratefully acknowledges funding
from the European Union’s Horizon 2020 research
and innovation programme under the Marie SkłodowskaCurie
Grant Agreement No. 643073 for providing opportunities
to discuss this work with members of the CRITICS
network

Description

This is the final version of the article. Available from American Physical Society via the DOI in this record.